P-Adic Neural Networks

Abstract

In recent years there has been an enormous development of the applications of neural networks for solving practical problems. Neural networks take their name and structure from the neuronal system of animals and humans. They can be described as a unit with an input and corresponding output. The input is taken from any time series composed by vectors of n variables containing the results of some experiment at time t and the output might be some state variable which summarizes the conclusion of the experiment at time t + 1 represented by a vector of n variables. The transformation which associates the output to a given input is defined in analogy with the behavior of the real neurones. For example the simplest structure is with one unit of m intermediate neurones while the input is described by n neurones and the output by q neurones. The input neurones are connected by synaptic weights w ¹ _ij , i = 1,...,m, j = 1,...,n to the neurones of the inner layer and the neurones of the inner layer are connected to the neurones of the output layer by other synaptic weights w ² _li , l = 1,..., q, i = 1,..., m. The input vector ξ _j , j = 1,..., n is transformed into a vector of outputs h _i , i = 1,..., m of the intermediate level by the law defined by the weightsw ¹ _ij :

$$ {h_i} = f(\sum\limits_{j = 1}^n {w_{ij}^1} {\xi _j}) $$

, where f is the transfer function of the neuron which has the simple form

$$ f(x) = \frac{1}{{1 + \exp ( - \lambda x)}},x \in R $$

withλ > 0 a given constant. The output of the network is the vector γl, l = 1,...q given by

$$ \gamma l = f(\sum\limits_{i = 1}^m {w_{li}^2} {h_i}) $$